One of the things I remember most about middle school math class is that I went through it in a perpetual state of disorganization. During one particularly bad spell, I lost two calculators within a week. The loss, and the reaction of my parents, drove me to try to fix the problem once and for all. My plan was simple: buy an expensive calculator with the hope that it’d serve as an incentive to keep track of my stuff. The next weekend, I took weeks of allowance money to the local Service Merchandise and bought a new HP-11C pocket calculator. Almost 30 years later, I still have both the calculator and a fascination for its unusual Reverse Polish Notation (RPN) user interface. Over those decades, I’ve also found out that RPN provides a good way to explore a number of fundamental ideas in the field of software design.

If you haven’t used an RPN calculator, your first attempt will probably be a confusing experience. Unlike most calculators, The HP didn’t have an ‘=’ key. Instead, it had a large button down the center labeled ‘Enter’, which pushed the most recently keyed number onto an internal four-level stack. The mathematical operations then worked against this stack. The ‘+’ key popped off two numbers, added them, and pushed the result back onto the stack. To add 2 and 3, you’d make the following keystrokes: [2][ENTER][3][+].

In great detail:

[2] – Begin entering the number ‘2’ into the top level of the stack.

[ENTER] – Duplicate the number ‘2’ on the top of the stack so it’s in the top two levels.

[3] – Begin entering the number ‘3’ into the top level of the stack, replacing one of the copies of the number ‘2’.

[+] – Pop off the top two levels of the stack, pushing back the sum of the two numbers.

Because the display shows the top level of the stack, it shows the answer (5) as soon as the user presses [+]. Because the answer, 5, is on the top of the stack, it’s also immediately ready to be used as an input to another calculation. This last bit is why RPN calculators can be so compelling to use: they make it easy to start with a small calculation and extend it into something larger. As long as a number is on the stack, it can be used in a calculation; The origin of the number doesn’t matter to the way it can be used. In computer science terms, this is the beginning of Referential Transparency. The FORTH programming language builds on this foundation, extending the basic tenets of RPN into a complete programming language. RPN combined with functional decomposition gives the language a great deal of expressive power, but due to the simplicity of stacks a small FORTH can be implemented in a very small amount of memory.